Final answer:
The correct form of the partial fraction decomposition for the expression (7x+18) / (x^2+9) is (Ax + B)/(x^2+9), since the denominator is a quadratic with no real roots and can't be split into real linear factors.
Step-by-step explanation:
The correct form of the partial fraction decomposition for the expression (7x+18) / (x^2+9) is not directly given in options a-d, but we can determine it based on the denominator. Since the denominator x^2+9 is a quadratic with no real roots, the partial fraction decomposition must take the form of (Ax+B)/(x^2+9). This is because partial fraction decomposition splits the fraction into simpler parts and since x^2+9 cannot be factored into real linear factors, we leave it as is, only factoring out an x term if it were present in the numerator which is not the case.
Options a and b are incorrect because they Suggest decomposing into linear or repeated linear factors which don't match the original denominator. Option c mistakenly includes C/9x indicating a potential repeated linear factor of x which is not present in x^2+9. Therefore, by process of elimination and understanding of partial fractions, we must adjust option d to (Ax + B)/x^2 + C/(x^2+9), but since C/(x^2+9) would be redundant, we simply use (Ax + B)/(x^2+9).